Subgroups — C₁₇×#8961 #16866705

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2975	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1479	C₂	1	2	yes	maximal, complemented
4	C₁₇×#1460	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1276	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#436	—	2	4	no	—
10	C₁₇×#431	C₂²	1	2	yes	—
11	C₁₇×#431	C₂²	1	2	yes	—
12	C₁₇×#407	—	2	2	no	—
13	C₁₇×#391	C₄	1	1	yes	characteristic
14	C₁₇×#390	C₄	1	1	yes	characteristic
15	C₁₇×#388	—	2	2	no	—
16	C₁₇×#382	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₁₇	—	2	2	no	—
18	C₄³×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×Q₈⋊C₈	—	2	4	no	—
20	C₁₇×#323	—	2	2	no	—
21	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₇	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₇	—	2	4	no	—
25	C₁₇×#124	—	2	2	no	—
26	C₁₇×#124	—	2	2	no	—
27	C₄×Q₈×C₁₇	—	2	2	no	—
28	C₄×Q₈×C₁₇	—	2	2	no	—
29	C₄×D₄×C₁₇	—	2	2	no	—
30	C₄×D₄×C₁₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
37	C₂×C₄²×C₁₇	—	2	1	no	—
38	C₂×C₄²×C₁₇	—	2	1	no	—
39	C₂×C₄²×C₁₇	—	2	1	no	—
40	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	—	2	2	no	—
44	C₁₇×C₄⋊C₈	D₄	1	4	yes	—
45	C₁₇×C₂²⋊C₈	—	2	4	no	—
46	C₁₇×C₂²⋊C₈	—	2	2	no	—
47	C₁₇×C₂²⋊C₈	—	2	2	no	—
48	C₄×C₈×C₁₇	—	2	2	no	—
49	C₄×C₈×C₁₇	—	2	2	no	—
50	C₁₇×#95	—	2	2	no	—
51	C₁₇×#95	—	2	2	no	—
52	C₂⁴×C₁₇	C₂²⋊C₄	1	2	yes	—
53	C₂×Q₈×C₁₇	—	2	1	no	—
54	C₂×Q₈×C₁₇	—	2	2	no	—
55	C₂×Q₈×C₁₇	—	2	2	no	—
56	C₂×Q₈×C₁₇	M₁₆	1	2	yes	complemented
57	C₂×D₄×C₁₇	—	2	2	no	—
58	C₂×D₄×C₁₇	—	2	1	no	—
59	C₂×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	2	no	—
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	2	no	—
65	C₂²×C₄×C₁₇	—	2	2	no	—
66	C₂²×C₄×C₁₇	—	2	1	no	—
67	C₂²×C₄×C₁₇	—	4	4	no	—
68	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₁₇	—	8	4	no	—
73	C₂×C₈×C₁₇	—	2	4	no	—
74	C₂×C₈×C₁₇	—	4	2	no	—
75	C₄⋊C₄×C₁₇	—	2	2	no	—
76	C₄⋊C₄×C₁₇	—	2	2	no	—
77	C₄⋊C₄×C₁₇	—	4	2	no	—
78	C₂²⋊C₄×C₁₇	—	2	8	no	—
79	C₂²⋊C₄×C₁₇	—	2	4	no	—
80	C₂²⋊C₄×C₁₇	—	4	4	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	—	2	2	no	—
83	C₄²×C₁₇	—	4	2	no	—
84	C₄²×C₁₇	—	4	2	no	—
85	C₄²×C₁₇	—	4	1	no	—
86	C₄²×C₁₇	—	4	1	no	—
87	C₄²×C₁₇	M₁₆	1	2	yes	complemented
88	C₄²×C₁₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₇	#41	1	2	yes	—
90	#8961	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₁₇	—	2	2	no	—
92	C₂³×C₁₇	—	2	2	no	—
93	C₂³×C₁₇	—	2	2	no	—
94	C₂³×C₁₇	—	4	2	no	—
95	C₂³×C₁₇	—	4	2	no	—
96	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	Q₈×C₁₇	—	2	4	no	—
98	Q₈×C₁₇	—	4	2	no	—
99	D₄×C₁₇	—	4	2	no	—
100	D₄×C₁₇	—	4	2	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	4	1	no	—
107	C₂×C₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	—	4	1	no	—
110	C₂×C₄×C₁₇	—	4	1	no	—
111	C₂×C₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	4	no	—
114	C₂×C₄×C₁₇	—	4	2	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	—	4	1	no	—
117	C₂×C₄×C₁₇	—	8	2	no	—
118	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₇	#121	1	2	yes	—
121	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₇	#131	1	2	yes	—
125	C₈×C₁₇	—	4	4	no	—
126	C₈×C₁₇	—	8	2	no	—
127	#2975	C₂×C₁₇	1	1	yes	characteristic
128	C₄×#124	C₂×C₁₇	1	1	yes	characteristic
129	#1479	C₂×C₁₇	1	2	yes	complemented
130	#1460	C₂×C₁₇	1	1	yes	characteristic
131	#1276	C₂×C₁₇	1	2	yes	complemented
132	C₂²×C₁₇	—	2	1	no	—
133	C₂²×C₁₇	—	2	1	no	—
134	C₂²×C₁₇	—	4	4	no	—
135	C₂²×C₁₇	—	4	2	no	—
136	C₂²×C₁₇	—	4	2	no	—
137	C₂²×C₁₇	—	4	2	no	—
138	C₂²×C₁₇	—	8	2	no	—
139	C₂²×C₁₇	#386	1	1	yes	characteristic
140	C₂²×C₁₇	#407	1	1	yes	characteristic
141	C₂²×C₁₇	#408	1	1	yes	characteristic
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	4	2	no	—
145	C₄×C₁₇	—	4	2	no	—
146	C₄×C₁₇	—	4	2	no	—
147	C₄×C₁₇	—	4	2	no	—
148	C₄×C₁₇	—	8	1	no	—
149	C₄×C₁₇	#435	1	2	yes	—
150	C₄×C₁₇	#436	1	2	yes	—
151	C₄×C₁₇	#455	1	2	yes	—
152	C₂×#124	C₄×C₁₇	1	1	yes	characteristic
153	C₂×C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
154	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
155	#436	—	2	4	no	—
156	#431	C₂²×C₁₇	1	2	yes	—
157	#431	C₂²×C₁₇	1	2	yes	—
158	#407	—	2	2	no	—
159	#391	C₄×C₁₇	1	1	yes	characteristic
160	#390	C₄×C₁₇	1	1	yes	characteristic
161	#388	—	2	2	no	—
162	#382	C₄×C₁₇	1	1	yes	characteristic
163	C₄×C₂²⋊C₄	—	2	2	no	—
164	C₄³	C₂²×C₁₇	1	1	yes	characteristic
165	Q₈⋊C₈	—	2	4	no	—
166	#323	—	2	2	no	—
167	C₂×C₁₇	—	2	1	no	—
168	C₂×C₁₇	—	2	1	no	—
169	C₂×C₁₇	—	8	2	no	—
170	C₂×C₁₇	#1870	1	1	yes	characteristic
171	C₂×C₁₇	#1884	1	1	yes	characteristic
172	C₂×C₁₇	#1885	1	1	yes	characteristic
173	C₂²×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
174	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₁₇	1	2	yes	—
176	C₂×M₁₆	—	2	4	no	—
177	#124	—	2	2	no	—
178	#124	—	2	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	D₄×C₁₇	1	2	yes	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	D₄×C₁₇	1	4	yes	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₁₇	#8961	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂²⋊C₄×C₁₇	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	M₁₆×C₁₇	1	2	yes	complemented
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
225	M₁₆	—	8	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	M₁₆×C₁₇	1	2	yes	complemented
241	C₄²	C₂×D₄×C₁₇	1	2	yes	—
242	C₄²	#41×C₁₇	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
271	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
272	C₂×C₄	C₁₇×#121	1	2	yes	—
273	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
274	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
275	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
276	C₂×C₄	C₁₇×#131	1	2	yes	—
277	C₈	—	4	4	no	—
278	C₈	—	8	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	C₁₇×#386	1	1	yes	characteristic
287	C₂²	C₁₇×#407	1	1	yes	characteristic
288	C₂²	C₁₇×#408	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	C₁₇×#435	1	2	yes	—
297	C₄	C₁₇×#436	1	2	yes	—
298	C₄	C₁₇×#455	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₁₇×#1870	1	1	yes	characteristic
303	C₂	C₁₇×#1884	1	1	yes	characteristic
304	C₂	C₁₇×#1885	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#386
prime radical	C₂²×D₄×C₁₇	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₁₇×#1479	C₂	semidirect
5. C₁₇×#1276	C₂	semidirect
56. C₂×Q₈×C₁₇	M₁₆	semidirect
87. C₄²×C₁₇	M₁₆	semidirect
90. #8961	C₁₇	direct (N × H)
129. #1479	C₂×C₁₇	semidirect
131. #1276	C₂×C₁₇	semidirect
204. C₁₇	#8961	direct (N × H)
209. C₂×Q₈	M₁₆×C₁₇	semidirect
240. C₄²	M₁₆×C₁₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₇×#2975	2
C₄×C₁₇×#124	2
C₁₇×#1479	2
C₁₇×#1460	2
C₁₇×#1276	2
#8961	17

Subgroups — C17×#8961 #16866705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#8961 #16866705